Rates of convergence for density estimation with generative adversarial networks
This provides theoretical guarantees for GANs in nonparametric density estimation, addressing a foundational gap in understanding their statistical performance.
The authors tackled the problem of non-asymptotic convergence rates for density estimation using vanilla GANs, proving an oracle inequality that yields a JS-divergence decay rate of (log n / n)^{2β/(2β+d)}, which matches minimax optimal rates up to logarithmic factors.
In this work we undertake a thorough study of the non-asymptotic properties of the vanilla generative adversarial networks (GANs). We prove an oracle inequality for the Jensen-Shannon (JS) divergence between the underlying density $\mathsf{p}^*$ and the GAN estimate with a significantly better statistical error term compared to the previously known results. The advantage of our bound becomes clear in application to nonparametric density estimation. We show that the JS-divergence between the GAN estimate and $\mathsf{p}^*$ decays as fast as $(\log{n}/n)^{2β/(2β+ d)}$, where $n$ is the sample size and $β$ determines the smoothness of $\mathsf{p}^*$. This rate of convergence coincides (up to logarithmic factors) with minimax optimal for the considered class of densities.